Higgo James wrote:
>
> But even such an 'identical' human has different spatial co-ordinates and is
> therefore different, no?
>
Umm, no! "Space" is an emergent phenomenon as well. What could you
possibly mean that it has a different "spatial coordinate"?
> > -----Original Message-----
> > From: Christopher Maloney [SMTP:dude.domain.name.hidden]
> > Sent: Wednesday, July 14, 1999 6:25 AM
> > To: everything-list
> > Subject: Re: Cardinality of the MW
> >
> > Let me add my information to this confusing brew:
> >
> > hal.domain.name.hidden wrote:
> > >
> > > Russell Standish, <R.Standish.domain.name.hidden>, writes:
> > > > My memory is fading somewhat about transfinite cardinal
> > > > numbers. However, it seems to me that c \leq \aleph_1. \aleph_1 is the
> > > > cardinality of the set of all sets of cardinalilty \leq\aleph_0. Since
> > c
> > > > is the cardinality of the set of all subsets of N, which is a subset
> > > > of the set of all sets of cardinality \leq\aleph_0.
> >
> > This is wrong, from what I know. I agree with Hal below that aleph-1
> > is defined to be the "next" cardinal after aleph-0. That is, by
> > definition, there is no transfinite number with cardinality > aleph-0
> > and < aleph-1.
> >
> > > >
> > > > What has never been proven is that c=\aleph_1, although it is widely
> > > > suspected.
> >
> > In fact it has been shown that c == aleph-1 is not provable by the
> > axioms of Zermelo-Fraenkel set theory. This is known as the
> > continuum hypothesis (CH). CH is also not disprovable,
> > which means that it is independent of those axioms. Thus it is
> > possible to construct set theories which assume that ~CH, and these
> > are known as non-Cantorian set theories.
> >
> > On the other hand, in standard set theory, assuming the CH does no
> > harm.
> >
> > [More below]
> >
> > >
> > > This is pretty much over my head. As I understand it aleph 1 is
> > > defined to be the next cardinal number after aleph 0, and can be
> > > shown to be the cardinality of the set of all countable ordinals
> > > (1...w...w+1...2w...w^2...). Since the elements of this set are all
> > > ordered sets (i.e. ordinals), while the subsets of N don't have an
> > > ordering requirement, this gives more flexibility to c and so you can't
> > > compare them as simply as you have shown here.
> > >
> > > See http://www.ii.com/math/ch/ for a detailed discussion of these
> > > matters.
> > >
> > > Actually on further thought I think I was wrong to suggest that the
> > number
> > > of TM programs is c, since that would allow for infinite length
> > programs,
> > > which is perhaps outside the spirit of a TM. If we require only finite
> > > length programs then the number of TMs is aleph 0 since we can enumerate
> > > all the programs, and that would be the number of universes as well.
> > > Not so many after all.
> > >
> > > Hal
> >
> > I think you are right that the cardinality of the set of all programs
> > is aleph-0.
> >
> > But neither of you (nor anyone else) has addressed my reason for
> > conjecturing that the set of branches in our structure must be
> > aleph-(aleph-0), which is based on the SSA.
> >
> > To tell you the truth, I'm certainly not convinced of it, but I
> > think it's worth considering. To discard the conclusion, I would
> > think that you'd have to assume "the identity of indiscernables".
> > My reasoning is illustrated if you only assume, for the moment,
> > that some observable (say, x) can take a continuum of possible
> > values when measured. Forget about the Plank length, for now.
> > That would mean that the set of all possible humans would have
> > cardinality c (at least). Thus it would be impossible to map
> > that set onto the set of all programs.
> >
> > But if you believe in the computationalist hypothesis, then you'd
> > have to assume that at some point, a simulation of a human becomes
> > "close enough" to be identical. That is (to oversimplify) when
> > each particle is simulated to within a Plank length, then the
> > simulation becomes indiscernable from the original, and thus
> > identical. If this is true, then I would no longer expect the
> > physical laws to give rise to ever-increasing cardinality of
> > universes, since that could never increase the cardinality of
> > the set of humans past aleph-0, anyway.
> >
> >
> >
> > --
> > Chris Maloney
> > http://www.chrismaloney.com
> >
> > "Knowledge is good"
> > -- Emil Faber
--
Chris Maloney
http://www.chrismaloney.com
"Knowledge is good"
-- Emil Faber
Received on Fri Jul 16 1999 - 20:57:58 PDT